We are given the equation:

$14(0.26)^x = 10$

We want to solve for $x$. Here are the steps:

Step 1: Isolate the exponential term

Divide both sides of the equation by 14 to isolate $(0.26)^x$:

$(0.26)^x = \frac{10}{14} = 0.7143$

Step 2: Take the natural logarithm of both sides

To solve for $x$, apply the natural logarithm (ln) to both sides:

$\ln((0.26)^x) = \ln(0.7143)$

Step 3: Apply the logarithm power rule

Using the property of logarithms $\ln(a^b) = b \ln(a)$, we simplify the left side:

$x \ln(0.26) = \ln(0.7143)$

Step 4: Solve for $x$

Now divide both sides by $\ln(0.26)$:

$x = \frac{\ln(0.7143)}{\ln(0.26)}$

Using a calculator to evaluate the logarithms:

$\ln(0.7143) \approx -0.3365 \quad \text{and} \quad \ln(0.26) \approx -1.3471$

Thus,

$x = \frac{-0.3365}{-1.3471} \approx 0.2498$

Final answer:

$x \approx 0.25$

Would you like any further details on these steps?

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Here are some related questions to expand your understanding:

1. How do logarithmic properties simplify solving exponential equations?
2. What if the base of the exponent was larger than 1? How would it affect the solution?
3. Can this problem be solved using logarithms of a different base, like log base 10?
4. How does changing the 14 in the equation to another number affect the solution?
5. How can you verify the solution graphically?

Tip: Always check your answer by substituting it back into the original equation to ensure correctness.